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\title{应用回归分析第1章：回归分析概述}
\author{HXQ ET AL}

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\begin{frame}
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\begin{frame}{第1章目录 }

\begin{enumerate}

\item[1.1.] 变量间的相关关系
\item[1.2.] 回归的思想与名称来源
\item[1.3.] 回归分析的主要内容与一般模型
\item[1.4.] 回归分析应用与发展简评

\end{enumerate}

\end{frame}

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\begin{frame}{1.1.1. 变量之间的确定关系}

\begin{itemize}
\item {\color{red} 问题：举例说明变量之间的确定关系，即函数关系。}

\item 解答：

    \begin{itemize}
    \item 例子1：记 $x$ 为保险公司承保的汽车数量，记 $y$ 为汽车承保的收入，设每辆汽车的保费为1000元，则有 $y=1000x$. 这是一个确定的关系。
    \item 例子2：设银行的一年期存款利率为2.55\%, 设 $x$ 为存入的本金，设 $y$ 为一年到期的本金加利息。则有
    $y=x+0.0255x$. 
    \end{itemize}

\end{itemize}

\end{frame}

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\begin{frame}{1.1.2. 变量之间的不确定关系}

\begin{itemize}
\item  {\color{red} 问题：举例说明变量之间的不确定关系。}

\item 解答：
    \begin{itemize}
    \item 设 $x$ 是居民收入，设 $y$ 是某种高档消费品的销售量。一般来讲，居民收入高，这种消费品的销售量就大。但是销售量不是完全由收入决定的，它还受到其它各种因素的影响。
    \item 粮食的产量 $Y$ 在较大程度上与施肥量 $x$ 由密切关系。但也不是完全确定的。
    \item 居民的储蓄额 $y$ 与居民的收入 $x$ 之间也是一种不确定的关系。
    \end{itemize}

\end{itemize}

\end{frame}

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\begin{frame}{1.1.3. 统计关系或相关关系}

\begin{itemize}
\item  {\color{red} 问题：什么是变量间的统计关系或相关关系？}

\item 解答：变量 $x$ 与 $y$ 之间有一定的关系，但是有没有密切到可以通过 $x$ 唯一确定 $y$ 的程度，
这种具有密切关联又不能由一个或一些变量唯一确定另外一个变量的关系称为统计关系或相关关系。

\item 关于统计关系的研究，有两个分支：
\begin{itemize}
\item 回归分析。
\item 相关分析。
\end{itemize}

\end{itemize}

\end{frame}

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\begin{frame}{1.1.4. 回归分析和相关分析的区别}

\begin{itemize}
\item 变量的地位有区别。
\begin{itemize}
\item 回归分析中，有自变量和因变量。
\item 相关分析中，各变量的地位是平等的。
\end{itemize}

\item 变量的随机性有区别。
\begin{itemize}
\item 回归分析中，自变量一般不当作随机变量，因变量当作随机变量。
\item 相关分析中，各变量都是随机变量。
\end{itemize}

\item 目的有区别。
\begin{itemize}
\item 回归分析的目的是了解自变量对因变量的影响，进行预测和控制。
\item 相关分析的主要目的是研究它们之间的密切程度。
\end{itemize}



\end{itemize}

\end{frame}

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\begin{frame}{1.1.5. 思考题1}

\begin{itemize}
\item  {\color{red} 问题：变量之间的统计关系和函数关系的区别是什么？}

\item 解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.1.6. 思考题2}

\begin{itemize}
\item  {\color{red} 问题：回归分析和相关分析的区别和联系是什么？}

\item 解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.2.1. 回归分析的基本思路、基本任务}

\begin{itemize}
\item[1.]  {\color{red} 问题：回归分析的基本思路是什么？}

\item 解答：当给定变量 $x$ 的值，随机变量 $y$ 的值不能确定，但是我们可以研究{\color{blue}条件期望}
\( f(x)=\text{E}(y\mid x),\)
这是一个确定的函数，称为随机变量 $y$ 对 $x$ 的均值回归函数。所以基本思路是使用条件期望这个概念。

\item[2.]  {\color{red} 问题：回归分析的基本任务是什么？}

\item 解答：给定变量 $(x,y)$ 的一些观测值，称为样本观测值：
\[ (x_1,y_1), (x_2,y_2), \cdots, (x_n,y_n),\]
基本任务有下述几个：

\begin{enumerate}
\item[(1)] 找出均值回归函数 $y=f(x)$.
\item[(2)]  给定变量 $x$ 的一个新的值，对变量 $y$ 的值作出预测。 
\end{enumerate}


\end{itemize}

\end{frame}

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\begin{frame}{1.2.2. 回归的名称由来}

\begin{itemize}
\item 回归分析的基本思想和方法来自英国统计学家高尔顿 (F. Galton, 1822-1911).

\item 高尔顿和皮尔逊(K. Pearson, 1856-1936) 在研究父母身高和子女身高的问题上，观察了1078对夫妇，以父母的平均身高为自变量 $x$, 一个成年儿子的身高为 $y$, 发现父母的平均身高增加1个单位，儿子的身高平均增加0.516个单位。

\item 为描述这种子代的身高有回到同龄人的平均高度的趋势，高尔顿使用了“回归”这个词。

\end{itemize}

\end{frame}

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\begin{frame}{1.2.3. 高尔顿的照片与优生学研究}

\begin{center}
\includegraphics[height=0.7\textheight, width=0.9\textwidth]{charles-darwins-cousin.jpg}
\end{center}

\end{frame}

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\begin{frame}{1.2.4. 高尔顿研究的身高数据}

\begin{center}
\includegraphics[height=0.7\textheight, width=0.9\textwidth]{galton-rtm-data.png}
\end{center}

\end{frame}

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\begin{frame}{1.2.5. 思考题1}

\begin{itemize}
\item  {\color{red} 问题：高尔顿的数据中，子代的身高相对于父代的身高，其发展趋势近乎一条直线吗？}

\item 解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.2.6. 思考题2}

\begin{itemize}
\item  {\color{red} 问题：回归分析的基本思想是什么？}

\item 解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.3.1. 回归分析的主要内容}

\begin{itemize}
\item 线性回归。
\item 回归诊断。
\item 回归变量的选择。
\item 参数估计方法的改进。
\item 非线性回归。
\item 含有定性变量的回归。
\end{itemize}

\end{frame}

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\begin{frame}{1.3.2. 回归诊断}

\begin{itemize}
\item  {\color{red} 问题：解释一下什么是回归诊断。}

\item 解答：
\begin{itemize}
\item 讨论如何从数据推断回归模型基本假设的合理性。
\item 当基本假设不成立时，如何对模型进行修正。
\item 判定回归方程拟合的效果。
\item 选择回归函数的形式。
\end{itemize}
\end{itemize}
\end{frame}

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\begin{frame}{1.3.3. 回归模型的一般形式}

\begin{itemize}
\item  {\color{red} 问题：回归模型的一般形式是什么？}

\item 解答：$y=f(x_1,x_2,\cdots,x_p)+\varepsilon$. 

\item 一些名词：
\begin{itemize}
\item 自变量 $x_1,x_2,\cdots,x_p$: 也称解释变量，内生变量。
\item 因变量 $y$: 也称被解释变量，外生变量。
\item 误差项 $\varepsilon$: 是一个随机变量，概括其它没有考虑到的因素和观测误差。
\end{itemize}

\item 回归模型的一般形式表达了自变量和因变量之间的相关关系既有联系又不确定的特点。

\end{itemize}

\end{frame}

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\begin{frame}{1.3.4. 线性回归模型的一般形式}

\begin{itemize}
\item  {\color{red} 问题：写出古典线性回归模型的一般形式。}

\item 解答：理论方程是 $y=\beta_0+\beta_1x_1+\beta_2x_2+\cdots+\beta_px_p+\varepsilon$.\\
代入观测数据后，得到一组方程：
\[ y_i=\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+\cdots+\beta_px_{ip}+\varepsilon_i,\,\, 1\le i\le n. \]

\item {\color{red} 问题：写出古典线性回归模型的基本假设。}

\item 解答：
\begin{enumerate}
\item 自变量不是随机变量，误差项和因变量是随机变量。
\item 误差项 $\varepsilon_i \,\, (1\le i\le n)$ 的均值为零，方差相等，两两不相关。
\item 更常用的假设是，误差项服从独立同分布的正态分布，均值为零。
\item 观测数据的个数多于自变量的个数，即 $n>p$. 
\end{enumerate}
\end{itemize}

\end{frame}

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\begin{frame}{1.3.5. 线性回归模型研究的问题}

\begin{itemize}
\item  {\color{red} 问题：线性回归模型通常研究哪些问题？}

\item 解答：
\begin{itemize}
\item 根据样本数据求出回归方程的系数和误差项的方差。
\item 验证基本假设是否符合，对系数和模型进行检验。
\item 根据回归方程进行预测和控制，理解实际问题中的变量之间的关系。
\end{itemize}
\end{itemize}
\end{frame}

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\begin{frame}{1.3.6. 思考题1}

\begin{itemize}
\item  {\color{red} 问题：回归模型中随机误差项的意义是什么？}

\item 解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.3.7. 思考题2}

\begin{itemize}
\item  {\color{red} 问题：如何理解线性回归模型的基本假设？}

\item 解答：


\end{itemize}

\end{frame}


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\begin{frame}{1.4.1. 简述回归模型的建立过程}

\begin{center}
\includegraphics[height=0.7\textheight, width=0.7\textwidth]{tu-1-3.jpeg}
\end{center}

\end{frame}

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\begin{frame}{1.4.2. 根据目的设置指标变量}

\begin{itemize}
\item 首先根据研究问题的目的设置因变量 $y$. 
\item 然后选取与因变量有统计关系的一些变量作为自变量。
\item 选择变量要与研究问题的领域专家合作。
\item 自变量并不是越多越好。

\end{itemize}

\end{frame}

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\begin{frame}{1.4.3. 收集整理数据}

\begin{itemize}
\item 时间序列数据是按照时间顺序排列的统计数据。
\item 时间序列数据要特别注意数据的可比性和数据的统计口径问题。如果不一致就要作调整。
\item 时间序列数据容易产生误差项的序列相关。
\item 横截面数据是在同一时间截面上的统计数据。
\item 横截面数据容易产生异方差性。
\item 为使模型参数估计有效，样本量要大于解释变量个数。M. Kendall 认为样本量应为解释变量个数的10倍。
\end{itemize}

\end{frame}

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\begin{frame}{1.4.4. 确定理论回归模型}

\begin{itemize}
\item 绘制 $(x_i,y_i),1\le i\le n$ 的散点图，确定函数形式，如下述候选函数，
\begin{eqnarray*}
y = a+bx,\hspace{0.5cm}
y = ae^{bx}
%\hspace{0.3cm}
%y=a+b\ln x. 
\end{eqnarray*}
\item 参考经济理论和数理经济学的结果，如CD生产函数：
\begin{eqnarray*}
y &=& AK^\alpha L^\beta U,\\
\ln y &=& \ln A + \alpha \ln K + \beta\ln L + \ln U.
\end{eqnarray*}

\item 模型的参数的正负符号经常可以根据实际问题事先确定。
%\item 
%\item 

\end{itemize}

\end{frame}

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\begin{frame}{1.4.5. 估计模型的参数}

\begin{itemize}
\item 估计未知参数的常用方法是最小二乘法。

\item 在最小二乘法的基础上，其它方法有岭回归、主成分回归、偏最小二乘估计等。

\item 参数估计一般用计算机软件完成，如 R, SPSS, SAS, JMP 等。

\end{itemize}

\end{frame}

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\begin{frame}{1.4.6. 模型的检验与改进}

\begin{itemize}
\item 模型检验包括统计意义上的检验和经济意义上的检验。

\item 统计检验：
\begin{itemize}
\item 回归方程是否显著？
\item 参数是否显著不等于零？
\item 回归模型的基本假设是否符合？
\end{itemize}
\item 根据经济含义，一般可以确定参数的正负号。
\item 改进模型的方法包括重新设置变量和改变回归函数形式等。

\end{itemize}
\end{frame}

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\begin{frame}{1.4.7. 回归模型的应用}

\begin{itemize}
\item 一个重要应用是经济变量的因素分析。

\item 根据模型反映出来的因果关系，对自变量进行控制，以达到控制因变量的效果。

\item 用回归模型进行经济预测。

\end{itemize}

\end{frame}

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\begin{frame}{1.4.8. 回归分析的发展}

\begin{itemize}
\item 回归分析广泛应用在经济领域。
\item 矩阵理论和计算机技术极大地促进了回归分析的应用。
\item 对于满足基本假设的回归模型，理论已经比较成熟。
\item 对于违背基本假设的回归模型，近年来仍有较多研究。
\item 回归分析理论仍是统计学家的研究课题。

\end{itemize}

\end{frame}

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\begin{frame}{1.4.9. 思考题1}

\begin{itemize}
\item  {\color{red} 问题：收集整理数据包括哪些内容？}

\item 解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.4.10. 思考题2}

\begin{itemize}
\item  {\color{red} 问题：为什么要对回归模型进行检验？}

\item 解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.4.10. 思考题3}

\begin{itemize}
\item  {\color{red} 问题：构造回归分析的理论模型的基本依据是什么？}

\item 解答：


\end{itemize}

\end{frame}


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\begin{frame}{1.5.1. 多项选择题}

\begin{itemize}
\item  {\color{red} 问题：第一章（回归分析概述）的学习目标有哪些？}
\begin{enumerate}
\item[A.]  了解回归分析的思想和名称的来源。
\item[B.]  掌握回归分析研究的范围。
\item[C.] 理解回归模型的一般形式和假设条件。
\item[D.] 掌握回归模型的建立过程。
\end{enumerate}

\item  {\color{red}解答：ABCD. }

\end{itemize}

\end{frame}


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\begin{frame}{1.5.2. 多项选择题}

\begin{itemize}
\item  {\color{red} 问题：下述例子中，变量之间是统计关系，而不是函数关系的，有哪些？}
%关于变量之间的统计关系和函数关系的异同点，下述说法中，正确的有哪些？
\begin{enumerate}
\item[A.]  设每辆汽车的保费为1000元，设 $x$ 为保险公司承保的汽车数量，设 $y$ 为汽车承保的收入。
\item[B.]  设银行的一年期存款利率为2.55\%, 设 $x$ 为存入的本金，设 $y$ 为一年到期的本金加利息。
\item[C.]  设 $x$ 是居民收入，设 $y$ 是某种高档消费品的销售量。
\item[D.]  设  $x$ 是每亩的施肥量，设 $y$ 是每亩的粮食的产量。
\end{enumerate}


\vspace{0.2cm}

\item  {\color{red}解答：CD.}

\end{itemize}



\end{frame}

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\begin{frame}{1.5.3. 多项选择题}

\begin{itemize}
\item  {\color{red} 问题：关于回归分析和相关分析的区别，下述说法中，正确的有哪些？}
\begin{enumerate}
\item[A.]  变量的地位有区别。回归分析中，有自变量和因变量；相关分析中，各变量的地位是平等的。
\item[B.]  变量的随机性有区别。回归分析中，自变量一般不当作随机变量，因变量当作随机变量；相关分析中，各变量都是随机变量。
\item[C.]  目的有区别。回归分析的目的是了解自变量对因变量的影响，进行预测和控制；相关分析的主要目的是研究它们之间的密切程度。
\item[D.]  研究变量的数目有区别。回归分析研究多个变量，相关分析研究两个变量之间的关系。
\end{enumerate}


\vspace{0.2cm}

\item  {\color{red}解答：ABC.}

\end{itemize}


\end{frame}

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\begin{frame}{1.5.4. 多项选择题}

\begin{itemize}
\item  {\color{red} 问题：关于回归分析的基本思路和任务，下述说法中，正确的有哪些？}
\begin{enumerate}
\item[A.]  为研究因素 $x$ 对变量 $y$ 的影响，将 $y$ 看作随机变量，研究均值回归函数 $\text{E}(y|x)$. 
\item[B.]  为研究因素 $x$ 对变量 $y$ 的影响，将 $x,y$ 看作随机变量，研究协方差 $\text{cov}(x,y)$. 
\item[C.]  基本任务之一是从样本观测值 \( (x_1,y_1), (x_2,y_2), \cdots, (x_n,y_n)\) 找出均值回归函数 $f(x)$.
\item[D.]  基本任务之一给定变量 $x$ 的一个新的值，对变量 $y$ 的值作出预测。
\end{enumerate}


\vspace{0.2cm}

\item  {\color{red}解答：ACD.}

\end{itemize}



\end{frame}

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\begin{frame}{1.5.5. 多项选择题}

\begin{itemize}
\item  {\color{red} 问题：关于回归的名称由来，下述说法中，正确的有哪些？}
\begin{enumerate}
\item[A.]  英国统计学家高尔顿和皮尔逊在研究父母身高和子女身高的问题上，观察了1078对夫妇。
\item[B.]  高尔顿以父母的平均身高为自变量 $x$, 一个成年儿子的身高为 $y$.  
\item[C.]  高尔顿发现父母的平均身高增加1个单位，儿子的身高平均增加0.516个单位。为描述这种子代的身高有回到同龄人的平均高度的趋势，高尔顿使用了“回归”这个词。
\item[D.]  高尔顿的数据中，子代的身高相对于父代的身高，其发展趋势近乎一条直线。
\end{enumerate}


\vspace{0.2cm}

\item  {\color{red}解答：ABC.}

\end{itemize}



\end{frame}

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\begin{frame}{1.5.6. 多项选择题}

\begin{itemize}
\item  {\color{red} 问题：关于回归模型的一般形式 $$y=f(x_1,x_2,\cdots,x_p)+\varepsilon,$$ 下述说法中，正确的有哪些？}
\begin{enumerate}
\item[A.]  自变量 $x_1,x_2,\cdots,x_p$ 也称解释变量或内生变量。
\item[B.]  因变量 $y$ 也称被解释变量或外生变量。
\item[C.]  误差项 $\varepsilon$ 是一个随机变量，概括其它没有考虑到的因素和观测误差。
\item[D.]  函数关系式 $f(x_1,x_2,\cdots,x_n)$ 是由统计学家预先确定的。
\end{enumerate}


\vspace{0.2cm}

\item  {\color{red}解答：ABC.}

\end{itemize}



\end{frame}

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\begin{frame}{1.5.7. 多项选择题}

\begin{itemize}
\item  {\color{red} 问题：如何理解古典线性回归模型 $$y=\beta_0+\beta_1x_1+\beta_2x_2+\cdots+\beta_px_p+\varepsilon$$ 的基本假设？} %下述说法中，正确的有哪些？
\begin{enumerate}
\item[A.]  自变量 $x_1,x_2,\cdots,x_p$ 不是随机变量，误差项 $\varepsilon$ 和因变量 $y$ 是随机变量。
\item[B.]  误差项 $\varepsilon_i \,\, (1\le i\le n)$ 的均值为零，方差相等，两两不相关。
\item[C.]  因变量的变化是由自变量引起的确定性的变化和其它因素引起的随机性的变化叠加而成。
\item[D.]  观测数据的个数多于自变量的个数，即 $n>p$. 
\end{enumerate}


\vspace{0.2cm}

\item  {\color{red}解答：ABCD. }

\end{itemize}



\end{frame}

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\begin{frame}{1.5.8. 多项选择题}

\begin{itemize}
\item  {\color{red} 问题：给了线性回归模型 $$y=\beta_0+\beta_1x_1+\beta_2x_2+\cdots+\beta_px_p+\varepsilon$$ 之后，需要研究哪些问题？} %下述说法中，正确的有哪些？
\begin{enumerate}
\item[A.]  根据样本数据求出回归方程的系数 $\beta_k$ 和误差项 $\varepsilon$ 的方差。
\item[B.]  验证基本假设是否符合，对系数和模型进行检验。
\item[C.]  根据回归方程进行预测和控制。
\item[D.]  理解实际问题中的变量之间的关系。
\end{enumerate}


\vspace{0.2cm}

\item  {\color{red}解答：ABCD. }

\end{itemize}



\end{frame}

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\begin{frame}{1.5.9. 多项选择题}

\begin{itemize}
\item  {\color{red} 问题：根据研究问题的目的，确定研究对象即因变量 $y$ 之后，如何设置自变量 $x_1,\cdots,x_p$ ? } %下述说法中，正确的有哪些？
\begin{enumerate}
\item[A.]  选取与因变量有统计关系的一些变量作为自变量，设置的自变量之间应该是不相关的。
\item[B.]  也可以使用几个因素组合而成的复合指标作为自变量。
\item[C.]  选择变量要与研究问题的领域专家合作。
\item[D.] 自变量总是越多越好，将所有与因变量相关的变量都包括进来。
\end{enumerate}


\vspace{0.2cm}

\item  {\color{red}解答：ABC.}

\end{itemize}


\end{frame}

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\begin{frame}{1.5.10. 多项选择题}

\begin{itemize}
\item  {\color{red} 问题：收集整理数据要注意哪些地方？}%下述说法中，正确的有哪些？
\begin{enumerate}
\item[A.]  时间序列数据是按照时间顺序排列的统计数据，容易产生误差项的序列相关。
\item[B.]  时间序列数据要特别注意数据的可比性和数据的统计口径问题，如果不一致就要作调整。
\item[C.] 横截面数据是在同一时间截面上的统计数据，这样的数据容易产生异方差性。
\item[D.] 为保证模型的有效性，M. Kendall 认为样本量应为解释变量个数的100倍。
\end{enumerate}


\vspace{0.2cm}

\item  {\color{red}解答：ABC. }

\end{itemize}



\end{frame}

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\begin{frame}{1.5.11. 多项选择题}

\begin{itemize}
\item  {\color{red} 问题：关于回归模型的检验，下述说法中，正确的有哪些？}
\begin{enumerate}
\item[A.]  回归诊断是讨论如何从数据推断回归模型基本假设的合理性。
\item[B.]  对回归模型进行统计检验，通常包括这几个问题：回归方程是否显著？参数是否显著不等于零？回归模型的基本假设是否符合？
\item[C.]  考察变量之间的经济含义，一般可以确定参数的正负号。
\item[D.]  检验通不过就需要改进模型，改进方法包括重新设置变量和改变回归函数形式等。
\end{enumerate}


\vspace{0.2cm}

\item  {\color{red}解答：ABCD. }

\end{itemize}



\end{frame}

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\begin{frame}{1.5.12. 多项选择题}

\begin{itemize}
\item  {\color{red} 问题：构造回归分析的理论模型的基本方法有哪些？}%下述说法中，正确的有哪些？
\begin{enumerate}
\item[A.]  绘制样本数据 $(x_i,y_i),1\le i\le n$ 的散点图，确定函数形式。
\item[B.]  参考经济理论和数理经济学的结果，参数的正负符号经常可以根据实际问题事先确定。
\item[C.]  使用随机数学的方法得出概率上的结论。
\item[D.]  将一些模型作为候选，使用计算机模拟，选择最好的模型。
\end{enumerate}

\vspace{0.2cm}

\item  {\color{red}解答：ABCD. }

\end{itemize}



\end{frame}

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\begin{frame}{1.5.13. 多项选择题}

\begin{itemize}
\item  {\color{red} 问题：关于回归分析的应用和发展，下述说法中，正确的有哪些？}
\begin{enumerate}
\item[A.]  回归分析广泛应用在经济领域。
\item[B.]  矩阵理论和计算机技术极大地促进了回归分析的应用。
\item[C.] 对于满足基本假设的回归模型，理论已经比较成熟。对于违背基本假设的回归模型，近年来仍有较多研究。
\item[D.] 回归分析理论现在仍是统计学家的研究课题。
\end{enumerate}

\vspace{0.2cm}

\item  {\color{red}解答：ABCD. }

\end{itemize}



\end{frame}

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\begin{frame}{1.5.14. 简答题 }

\begin{itemize}
\item  {\color{red} 问题：关于古典线性回归模型。}
\begin{enumerate}
\item[(1)]  写出古典线性回归模型的数学形式。
\item[(2)]  写出高斯-马尔可夫条件。
\item[(3)]  写出正态分布的假设条件。
\item[(4)]  解释回归模型中的随机误差项的意义。
\end{enumerate}


\item  {\color{red}解答：

\begin{enumerate}
\item[(1)]  古典线性回归模型的理论方程和样本方程分别是
\begin{eqnarray*}
y &=& \beta_0+\beta_1x_1+\beta_2x_2+\cdots+\beta_px_p+\varepsilon,\\
y_i &=& \beta_0+\beta_1x_{i1}+\beta_2x_{i2}+\cdots+\beta_px_{ip}+\varepsilon_i,\,\, 1\le i\le n.
\end{eqnarray*}

\end{enumerate}

}

\end{itemize}

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\begin{frame}{1.5.14. 简答题解答}

\begin{enumerate}

\item[(2)]  高斯-马尔可夫条件是指误差项 $\varepsilon_i \,\, (1\le i\le n)$ 的均值为零，方差相等，两两不相关。数学表达就是
\begin{eqnarray*}
\left\{\begin{array}{ll}
\text{E}(\varepsilon_i) = 0, \,\,\, \text{var}(\varepsilon_i) = \sigma^2, & i=1,2,\cdots n\\
\text{cov}(\varepsilon_i,\varepsilon_j) = 0, &  i\neq j, \,\, i,j=1,2,\cdots n
\end{array}\right.
\end{eqnarray*}

\item[(3)]  正态假设是指误差项服从独立同分布的正态分布，数学表达就是
\[ \varepsilon_i \overset{\text{iid}}{\sim} N(0,\sigma^2),\,\,\, 1\le i\le n. \]

\item[(4)]  误差项反映了没有选入自变量的其它因素对因变量的影响，反映了自变量的线性组合未能解释因变量的部分影响，也反映了测量的误差，等。
\end{enumerate}


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\end{document}



